Boolean Algebras

Roman Sikorski (Autor)

Buch | Softcover

X, 240 Seiten

2012 | 3. Softcover reprint of the original 3rd ed. 1969
Springer Berlin (Verlag)
978-3-642-85822-2 (ISBN)

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There are two aspects to the theory of Boolean algebras; the algebraic and the set-theoretical. A Boolean algebra can be considered as a special kind of algebraic ring, or as a generalization of the set-theoretical notion of a field of sets. Fundamental theorems in both of these directions are due to M. H. STONE, whose papers have opened a new era in the develop ment of this theory. This work treats the set-theoretical aspect, with little mention being made of the algebraic one. The book is composed of two chapters and an appendix. Chapter I is devoted to the study of Boolean algebras from the point of view of finite Boolean operations only; a greater part of its contents can be found in the books of BIRKHOFF [2J and HERMES [1]. Chapter II seems to be the first systematic study of Boolean algebras with infinite Boolean operations. To understand Chapters I and II it suffices only to know fundamental notions from general set theory and set-theoretical topology. No know ledge of lattice theory or of abstract algebra is presumed. Less familiar topological theorems are recalled, and only a few examples use more advanced topological means; but these may be omitted. All theorems in both chapters are given with full proofs.

I. Finite joins and meets.-
1. Definition of Boolean algebras.-
2. Some consequences of the axioms.-
3. Ideals and filters.-
4. Subalgebras.-
5. Homomorphisms, isomorphisms.-
6. Maximal ideals and filters.-
7. Reduced and perfect fields of sets.-
8. A fundamental representation theorem.-
9. Atoms.-
10. Quotient algebras.-
11. Induced homomorphisms between fields of sets.-
12. Theorems on extending to homomorphisms.-
13. Independent subalgebras. Products.-
14. Free Boolean algebras.-
15. Induced homomorphisms between quotient algebras.-
16. Direct unions.-
17. Connection with algebraic rings.- II. Infinite joins and meets.-
18. Definition.-
19. Algebraic properties of infinite joins and meets. (m, n)-distributivity..-
20. m-complete Boolean algebras.-
21. m-ideals and m-filters. Quotient algebras.-
22. m-homomorphisms. The interpretation in Stone spaces.-
23. m-subalgebras.-
24. Representations by m-fields of sets.-
25. Complete Boolean algebras.-
26. The field of all subsets of a set.-
27. The field of all Borel subsets of a metric space.-
28. Representation of quotient algebras as fields of sets.-
29. A fundamental representation theorem for Boolean ?-algebras. m-representability.-
30. Weak m-distributivity.-
31. Free Boolean m-algebras.-
32. Homomorphisms induced by point mappings.-
33. Theorems on extension of homomorphisms.-
34. Theorems on extending to homomorphisms.-
35. Completions and m-completions.-
36. Extensions of Boolean algebras.-
37. m-independent subalgebras. The field m-product.-
38. Boolean (m, n)-products.-
39. Relation to other algebras.-
40. Applications to mathematical logic. Classical calculi.-
41. Topology in Boolean algebras.Applications to non-classical logic.-
42. Applications to measure theory.-
43. Measurable functions and real homomorphisms.-
44. Measurable functions. Reduction to continuous functions.-
45. Applications to functional analysis.-
46. Applications to foundations of the theory of probability.-
47. Problems of effectivity.- List of symbols.- Author Index.

Erscheint lt. Verlag	9.4.2012
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Zusatzinfo	X, 240 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	391 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Algebra • Boolean algebra • boolescher Verband • Finite • Proof • set theory • Theorem • Topology
ISBN-10	3-642-85822-8 / 3642858228
ISBN-13	978-3-642-85822-2 / 9783642858222
Zustand	Neuware